On Subgradients of Spectral Functions

Let $F:\Sy{m}\rightarrow\Ret$ be a {\em spectral function} (i.e.\ $\Sy{m}$ is the space of $m\times m$ real symmetric matrices, $\forall O\in\Or{m},\forall X\in\Sy{m},\ F(OX{^tO})=F(X)$, where $\Or{m}$ is the orthogonal group and ${^tO}$ is the transpose of $O$). We associate to it the symmetric function $s_F:\R^m\rightarrow\Ret$ by restricting it to the subspace of diagonal matrices. In this work, on the one hand, we give a new, natural proof of the formula which binds the Fr\'echet subgradients of a spectral function $F$ and the Fr\'echet subgradients of the function $s_F$ (identical formulas follow for the subgradients and the horizon subgradients); on the other hand we deduce from the previous results and from convexity arguments that, in the general case, a similar formula holds for the Clarke subgradients.